A. Lim ⁡ X → 0 1 − 1 + X X \lim_{x \rightarrow 0} \frac{1-\sqrt{1+x}}{x} Lim X → 0 ​ X 1 − 1 + X ​ ​

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A Limit Problem: Evaluating the Indeterminate Form of lim⁡x→01−1+xx\lim_{x \rightarrow 0} \frac{1-\sqrt{1+x}}{x}

In mathematics, limits are a fundamental concept used to study the behavior of functions as the input values approach a specific point. The limit of a function as x approaches a certain value can be used to determine the function's behavior at that point. In this article, we will discuss the evaluation of the limit of the function 1−1+xx\frac{1-\sqrt{1+x}}{x} as x approaches 0.

The given limit problem is lim⁡x→01−1+xx\lim_{x \rightarrow 0} \frac{1-\sqrt{1+x}}{x}. This is an indeterminate form of type 00\frac{0}{0}, which means that the numerator and denominator both approach 0 as x approaches 0. To evaluate this limit, we need to use a suitable technique to simplify the expression and find its limit.

Rationalizing the Denominator

One way to simplify the expression is to rationalize the denominator by multiplying both the numerator and denominator by the conjugate of the numerator. The conjugate of 1−1+x1-\sqrt{1+x} is 1+1+x1+\sqrt{1+x}. Multiplying both the numerator and denominator by this conjugate, we get:

1−1+xx⋅1+1+x1+1+x=(1−1+x)(1+1+x)x(1+1+x)\frac{1-\sqrt{1+x}}{x} \cdot \frac{1+\sqrt{1+x}}{1+\sqrt{1+x}} = \frac{(1-\sqrt{1+x})(1+\sqrt{1+x})}{x(1+\sqrt{1+x})}

Simplifying the numerator, we get:

(1−1+x)(1+1+x)=1−(1+x)=−x(1-\sqrt{1+x})(1+\sqrt{1+x}) = 1 - (1+x) = -x

So, the expression becomes:

−xx(1+1+x)\frac{-x}{x(1+\sqrt{1+x})}

Simplifying the Expression

We can simplify the expression further by canceling out the common factor of x in the numerator and denominator:

−xx(1+1+x)=−11+1+x\frac{-x}{x(1+\sqrt{1+x})} = \frac{-1}{1+\sqrt{1+x}}

Evaluating the Limit

Now that we have simplified the expression, we can evaluate the limit as x approaches 0:

lim⁡x→0−11+1+x\lim_{x \rightarrow 0} \frac{-1}{1+\sqrt{1+x}}

As x approaches 0, the denominator approaches 1, and the numerator approaches -1. Therefore, the limit is:

lim⁡x→0−11+1+x=−11+1=−12\lim_{x \rightarrow 0} \frac{-1}{1+\sqrt{1+x}} = \frac{-1}{1+1} = \frac{-1}{2}

In this article, we discussed the evaluation of the limit of the function 1−1+xx\frac{1-\sqrt{1+x}}{x} as x approaches 0. We used the technique of rationalizing the denominator to simplify the expression and find its limit. The final answer is −12\frac{-1}{2}.

  • Limits of Functions: Limits are a fundamental concept in calculus used to study the behavior of functions as the input values approach a specific point.
  • Indeterminate Forms: Indeterminate forms are a type of limit problem where the numerator and denominator both approach 0 or infinity, making it difficult to evaluate the limit.
  • Rationalizing the Denominator: Rationalizing the denominator is a technique used to simplify expressions by multiplying both the numerator and denominator by the conjugate of the numerator.
  • Calculus: Calculus is a branch of mathematics that deals with the study of continuous change, including limits, derivatives, and integrals.
  • Mathematical Analysis: Mathematical analysis is a branch of mathematics that deals with the study of mathematical functions and their properties, including limits and derivatives.
  • Algebra: Algebra is a branch of mathematics that deals with the study of variables and their relationships, including equations and functions.
    A Limit Problem: Evaluating the Indeterminate Form of lim⁡x→01−1+xx\lim_{x \rightarrow 0} \frac{1-\sqrt{1+x}}{x} - Q&A

In our previous article, we discussed the evaluation of the limit of the function 1−1+xx\frac{1-\sqrt{1+x}}{x} as x approaches 0. We used the technique of rationalizing the denominator to simplify the expression and find its limit. In this article, we will provide a Q&A section to help clarify any doubts and provide further understanding of the concept.

Q: What is the concept of limits in mathematics?

A: Limits are a fundamental concept in mathematics used to study the behavior of functions as the input values approach a specific point. It helps us understand how a function behaves as the input values get arbitrarily close to a certain point.

Q: What is an indeterminate form?

A: An indeterminate form is a type of limit problem where the numerator and denominator both approach 0 or infinity, making it difficult to evaluate the limit. In the case of the given limit problem, it is an indeterminate form of type 00\frac{0}{0}.

Q: Why do we need to rationalize the denominator?

A: We need to rationalize the denominator to simplify the expression and make it easier to evaluate the limit. Rationalizing the denominator involves multiplying both the numerator and denominator by the conjugate of the numerator, which helps to eliminate the radical in the denominator.

Q: Can you explain the concept of conjugate in mathematics?

A: In mathematics, a conjugate is a pair of expressions that have the same terms but with opposite signs. For example, the conjugate of 1−1+x1-\sqrt{1+x} is 1+1+x1+\sqrt{1+x}. Multiplying a binomial by its conjugate eliminates the radical.

Q: How do we simplify the expression after rationalizing the denominator?

A: After rationalizing the denominator, we can simplify the expression by canceling out the common factor of x in the numerator and denominator. This helps to simplify the expression and make it easier to evaluate the limit.

Q: What is the final answer to the limit problem?

A: The final answer to the limit problem is −12\frac{-1}{2}.

Q: Can you provide more examples of limit problems?

A: Yes, here are a few more examples of limit problems:

  • lim⁡x→0x2−1x2+1\lim_{x \rightarrow 0} \frac{x^2-1}{x^2+1}
  • lim⁡x→01−cos⁡xx2\lim_{x \rightarrow 0} \frac{1-\cos x}{x^2}
  • lim⁡x→0sin⁡xx\lim_{x \rightarrow 0} \frac{\sin x}{x}

In this article, we provided a Q&A section to help clarify any doubts and provide further understanding of the concept of limits and indeterminate forms. We also provided a few more examples of limit problems to help reinforce the concept.

  • Limits of Functions: Limits are a fundamental concept in calculus used to study the behavior of functions as the input values approach a specific point.
  • Indeterminate Forms: Indeterminate forms are a type of limit problem where the numerator and denominator both approach 0 or infinity, making it difficult to evaluate the limit.
  • Rationalizing the Denominator: Rationalizing the denominator is a technique used to simplify expressions by multiplying both the numerator and denominator by the conjugate of the numerator.
  • Calculus: Calculus is a branch of mathematics that deals with the study of continuous change, including limits, derivatives, and integrals.
  • Mathematical Analysis: Mathematical analysis is a branch of mathematics that deals with the study of mathematical functions and their properties, including limits and derivatives.
  • Algebra: Algebra is a branch of mathematics that deals with the study of variables and their relationships, including equations and functions.